Originally Posted by

**Rozaline** I am having a lot of trouble attempting this problem. I do not know what is the best way to go about proving it.

Suppose sequence $\displaystyle a_n > 0$ and $\displaystyle b_n = a_n + \frac{1}{a_n}

$

Assume $\displaystyle a_n$ >= 1 for all n, and that $\displaystyle b_n$ converges. Prove that $\displaystyle a_n $ converges.

If it assumed that $\displaystyle b_n$ converges but only$\displaystyle a_n > 0$, it does not follow that $\displaystyle a_n$ converges. Find the example.